Theorem distgp 24080

Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
distgp.1	⊢ 𝐵 = (Base‘𝐺)
distgp.2	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
distgp	⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2		simpr 484	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3		distgp.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	3	fvexi 6852	. . . . 5 ⊢ 𝐵 ∈ V
5		distopon 22978	. . . . 5 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝒫 𝐵 ∈ (TopOn‘𝐵)
7	2, 6	eqeltrdi 2845	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
8		distgp.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝐺)
9	3, 8	istps 22915	. . 3 ⊢ (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
10	7, 9	sylibr 234	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
11		eqid 2737	. . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺)
12	3, 11	grpsubf 18992	. . . . 5 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵)
13	12	adantr 480	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵)
14	4, 4	xpex 7704	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
15	4, 14	elmap 8816	. . . 4 ⊢ ((-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)) ↔ (-g‘𝐺):(𝐵 × 𝐵)⟶𝐵)
16	13, 15	sylibr 234	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ (𝐵 ↑m (𝐵 × 𝐵)))
17	2, 2	oveq12d 7382	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
18		txdis 23613	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
19	4, 4, 18	mp2an 693	. . . . . 6 ⊢ (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
20	17, 19	eqtrdi 2788	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
21	20	oveq1d 7379	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
22		cndis 23272	. . . . 5 ⊢ (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵)))
23	14, 7, 22	sylancr 588	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵)))
24	21, 23	eqtrd 2772	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵 ↑m (𝐵 × 𝐵)))
25	16, 24	eleqtrrd 2840	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
26	8, 11	istgp2 24072	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
27	1, 10, 25, 26	syl3anbrc 1345	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  𝒫 cpw 4542   × cxp 5626  ⟶wf 6492  ‘cfv 6496  (class class class)co 7364   ↑_m cmap 8770  Basecbs 17176  TopOpenctopn 17381  Grpcgrp 18906  -_gcsg 18908  TopOnctopon 22891  TopSpctps 22913   Cn ccn 23205   ×_t ctx 23541  TopGrpctgp 24052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-1st 7939  df-2nd 7940  df-map 8772  df-0g 17401  df-topgen 17403  df-plusf 18604  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18909  df-minusg 18910  df-sbg 18911  df-top 22875  df-topon 22892  df-topsp 22914  df-bases 22927  df-cn 23208  df-cnp 23209  df-tx 23543  df-tmd 24053  df-tgp 24054

This theorem is referenced by: (None)